Rigid analytic space

Summary

A rigid analytic space is the analogue of a complex analytic space over a complete non-archimedean field such as the $p$-adic numbers. It is glued from affinoid spaces, the maximal spectra of quotients of algebras of convergent power series, along a Grothendieck topology of admissible opens. John Tate introduced the theory to describe elliptic curves with split multiplicative reduction as quotients of the multiplicative group. It underlies $p$-adic uniformization and $p$-adic Hodge theory.